Method and apparatus for rejecting the second harmonic current in an active converter with an unbalanced AC line voltage source

ABSTRACT

A method and apparatus for altering converter control as a function of the degree of unbalance in supply line voltages to substantially eliminate second harmonics on the supply lines caused by active control of the converter including identifying supply line peak amplitude values and using the peak values to identify potential second harmonics and then altering command values as a function of the potential second harmonic components.

CROSS-REFERENCE TO RELATED APPLICATIONS

Not applicable.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

BACKGROUND OF THE INVENTION

Power consuming clients that link to a utility power grid and draw powertherefrom have to limit the amount of disturbance they cause at the gridcoupling point so that other customers linked to the grid can rely on apower at the coupling point that has at least certain characteristics(e.g., a limited amount of harmonics, a limited amount of unbalance,etc.). To this end a series of regulations (e.g., IEEE 519) have beenadopted that specify grid linkage/power usage requirements.

AC power delivered to coupling points via grid lines (i.e., supplylines) is usually not in a condition that can be used by end users andtherefore the power at the coupling point must be converted so as tohave characteristics required by the end users. For instance, grid ACpower is often converted from AC to DC via a rectifier and then back toAC by an inverter where the amplitude and frequency is altered by theAC-DC-AC conversion and the resulting power is in a form useable topower end user equipment (e.g., motors, computers, office equipment,etc.).

To convert supply line AC power to DC power, the power conversionindustry has developed various converter topologies and methods. Forinstance, one common converter topology includes a six-pulse full waveconverter. Six pulse converters are advantageous because they arerelatively simple and inexpensive to construct. Unfortunately, six-pulseconverters have been known to generate high levels of harmonics (e.g.,fifth, seventh, third, eleventh, thirteenth, etc.) on linked supplylines which render these converters unusable under certain circumstancesor useable only if other conditioning hardware is used therewith. Here,the other conditioning hardware adds expense to the overall system.

Other converter types that overcome some of the shortcomings of the 6pulse type include a 12 pulse converter and an 18 pulse converter. Aswell known in the power conversion industry, 12 and 18 pulse convertersare able to reduce harmonic distortion when controlled in certain waysand when used to convert balanced supply line voltages. Unfortunately,when supply line voltages are unbalanced, it has been observed that 12and 18 pulse conversion can result in significant harmonic distortion.

Still one other converter type is generally referred to as an activeconverter where converter switching devices are actively controlled tofacilitate four quadrant operation (i.e., where the converter can beused in a bi-directional manner—as a converter from the grid to the DCbus or as an inverter from the bus to the grid). In addition to otheradvantages, active converters can reduce supply line harmonics whenlinked with balanced supply lines such that IEEE 519 standards are met.Unfortunately, it has been observed that when supply line voltages areunbalanced, active conversion can generate second harmonics that exceedtolerable levels.

In addition, when the supply line voltages are unbalanced, activeconversion often results in unbalanced current draw. Convertercomponents are usually rated for use with specific maximum or steadystate currents and therefore, where currents drawn are unbalanced, thephase of the conversion hardware carrying the highest current must beused to limit conversion rate. In other words, once the current throughone phase reaches the rated current level, the converter capacity mustbe limited to protect that phase despite the fact that the other twophases may have current levels far below the rated level.

Moreover, it has been observed that under certain circumstancesunbalanced supply line voltage causes increased voltage ripple on the DCbus (i.e., the link between the converter and the inverter in anAC-DC-AC conversion topology) which can cause increased heating and canshorten the useful life of conversion hardware components as well s theuseful life of inverter components linked to the DC bus.

Thus, it would be advantageous to have an AC-DC conversion configurationthat could simply and inexpensively maintain supply line harmonicsincluding the second harmonic to below tolerable threshold levels andthat could minimize DC bus voltage ripple even where supply linevoltages are unbalanced.

BRIEF SUMMARY OF THE INVENTION

Certain aspects commensurate in scope with the originally claimedinvention are set forth below. It should be understood that theseaspects are presented merely to provide the reader with a brief summaryof certain forms the invention might take and that these aspects are notintended to limit the scope of the invention. Indeed, the invention mayencompass a variety of aspects that may not be set forth below.

It has been recognized that, given certain reasonable assumptions,equations can be formulated to identify the amplitudes of supply linevoltages from RMS line-to-line voltages which can in turn be used toidentify supply line voltages that would result from active conversionof the unbalanced voltages. Once the second harmonic components of thesupply line voltages that would result from normal active conversionhave been identified, the identified second harmonic components can beused to modify converter command or control signals so that the secondharmonics that are actually generated are substantially minimized. Whenthe second harmonics in the supply line voltages are minimized, theconverter currents are more balanced and a conversion rate (i.e.,converter capacity) can be increased. In addition, when the supply linesecond harmonics are minimized, DC bus ripple is substantially reduced.

The compensation step can be performed in either a two phase synchronousdq frame of reference or in a three phase stationary frame of reference.The calculations in the two phase reference frame are more complex thanin the three phase reference frame and therefore, all other things beingequal, it may be advantageous to perform the process in a three phasereference frame. However, many existing controllers are programmed tooperate in the two phase reference frame and therefore, in many cases,two phase compensation may be advantageous.

Consistent with the above comments, the present invention includes amethod for use with a controller and a converter where the controlleractively controls the converter to convert three phase voltages on threesupply lines to a DC voltage across positive and negative DC buses, themethod comprising the steps of identifying the peak amplitudes of thethree phase supply line voltages, using the peak amplitudes to identifya second harmonic component that would be generated on the supply linesby the converter during normal operation due to unbalance in the peakamplitudes and altering control of the converter as a function of theidentified second harmonic.

In at least some embodiments the controller generates command voltagesto control the converter, the step of altering control of the converterincluding modifying the command voltages as a function of the identifiedsecond harmonic. Here, the step of identifying the peak amplitudes mayinclude sensing the RMS line-to-line voltages and using the RMSline-to-line voltages to identify the peak amplitudes.

In some cases the controller receives a DC reference voltage andcontrols the converter to cause the DC voltage across the DC buses totrack the reference voltage, the method further including the step ofusing the reference voltage to identify d and q-axis voltage differencevalues, the step of identifying the second harmonic components includingthe step of identifying d and q-axis components of the second harmonic,the step of modifying the command voltages including mathematicallycombining the difference values and the second harmonic components toidentify two phase d and q-axis command voltages.

The step of mathematically combining may include adding the d and q-axissecond harmonic components to the d and q-axis difference values.

The method may further include the step of using the peak amplitudes toidentify a DC offset and wherein the step of modifying the commandvoltages further includes adding the DC offset to the q-axis differencevalue along with the q-axis second harmonic component.

In some embodiments the peak amplitudes of the three supply linevoltages are a, b and c and wherein the step of identifying the d and qaxis second harmonic components includes using the peak amplitudes toidentify a two phase amplitude A_(min).

In some cases the method may further include the steps of identifyingthe frequency of the second harmonic of the supply line voltages andusing the frequency to identify a two phase supply voltage angle, thestep of identifying the second harmonic components including identifyingthe q-axis second harmonic component by multiplying value A_(min) by thesine of the voltage angle and identifying the d-axis second harmoniccomponent by multiplying value A_(min) by the cosine of the voltageangle.

In at least some embodiments the step of using the reference voltage toidentify d and q-axis voltage difference values includes using thereference voltage to identifying d and q-axis command currents,obtaining d and q-axis feedback currents, subtracting the d and q-axisfeedback currents from the d and q-axis command currents, respectively,and using the d and q-axis command currents to identify the voltagedifference values.

In some embodiments the controller receives a DC reference voltage andcontrols the converter to cause the DC voltage across the DC buses totrack the reference voltage, the method further including the step ofusing the reference voltage to identify first, second and third phasevoltage difference values, the step of identifying the second harmoniccomponents including the step of identifying first, second and thirdphase components of the second harmonic corresponding to the first,second and third supply lines, respectively, the step of modifying thecommand voltages including mathematically combining the differencevalues and the second harmonic components to identify first, second andthird command voltages.

The step of mathematically combining may include adding the first,second and third phase second harmonic components to the first, secondand third phase difference values. The method of claim 11 furtherincluding the step of identifying the frequency of the supply linevoltages and using the frequency to identify a supply voltage angle, thestep of identifying the second harmonic components including identifyingthe q-axis second harmonic component by multiplying peak values a, b andc by the sine of the supply voltage angle, the sine of the supplyvoltage angle less 120 degrees and the sine of the supply voltage angleplus 120 degrees, respectively.

The step of using the reference voltage to identify voltage differencevalues may include using the reference voltage to identifying first,second and third phase reference currents, obtaining first, second andthird phase feedback currents, subtracting the first, second and thirdphase feedback currents from the first, second and third referencecurrents to identify first, second and third command currents,respectively, and using the first, second and third phase commandcurrents to identify the voltage difference values.

The invention also includes a processor of controller programmed toperform the various processes and methods described above and hereafter.

The invention also contemplates a method for use with a controller and aconverter wherein the controller receives a reference voltage andgenerates first, second and third phase control voltages as a functionof the reference voltage, the converter receiving the first, second andthird phase control voltages and first, second and third phase linevoltages and converting the line voltages to a DC voltage acrosspositive and negative DC buses as a function of the control voltageswhere the line voltages may be unbalanced, the method for substantiallyreducing the second harmonics in the first, second and third phase linecurrents caused by drawing current from the lines when the line voltagesare unbalanced, the method comprising the steps of identifying first,second and third RMS line-to-line voltages, using the RMS line-to-linevoltages to identify peak phase voltage values, mathematically combiningthe peak phase voltage values and at least a derivative of the referencevoltage to generate the first, second and third phase command voltages;and using the first, second and third phase command voltages to controlthe converter.

At least some embodiments of the invention include a method for use witha controller and a converter where the controller actively controls theconverter to convert three phase voltages on three supply lines to a DCvoltage across positive and negative DC buses, the method comprising thesteps of identifying unbalance in the peak amplitudes of the three phasesupply line voltages and using the unbalance to alter control of theconverter to substantially eliminate generation of second harmonics onthe supply lines due to active converter control.

These and other objects, advantages and aspects of the invention willbecome apparent from the following description. In the description,reference is made to the accompanying drawings which form a part hereof,and in which there is shown a preferred embodiment of the invention.Such embodiment does not necessarily represent the full scope of theinvention and reference is made therefore, to the claims herein forinterpreting the scope of the invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The invention will hereafter be described with reference to theaccompanying drawings, wherein like reference numerals denote likeelements, and:

FIG. 1 is a schematic diagram of a prior art active converter electricalcircuit;

FIG. 2 is a phasor diagram illustrating the relationships betweenphasors in a stationary three phase system and phasors in a stationarytwo phase system;

FIG. 3 is schematic diagram illustrating a controller and a two phaseobject model according to at least some embodiments of the presentinvention;

FIG. 4 is a schematic diagram illustrating detailed components of thefeedforward voltage calculator of FIG. 3;

FIG. 5 is similar to FIG. 3, albeit illustrating a three phasecontroller and a three phase object model consistent with at least someembodiments of the present invention;

FIG. 6 is a schematic diagram illustrating detail of at least oneexemplary feedforward voltage calculator that may be used in theconfiguration of FIG. 5;

FIG. 7 is a graph illustrating RMS current waveforms generated using aconventional dq current loop controller where supply line voltages werebalanced prior to time 1.0 and unbalanced thereafter;

FIG. 8 is similar to FIG. 7, albeit illustrating waveforms generatedusing a controller operated in accordance with the present invention;

FIG. 9 is a graph illustrating steady state current waveforms generatedusing a conventional dq current loop controller;

FIG. 10 is similar to FIG. 9, albeit illustrating steady state currentwaveforms generated using a controller operated in a manner consistentwith the present invention;

FIG. 11 is a graph illustrating a DC bus voltage waveform generatedusing a conventional dq current loop controller where supply linevoltages were balanced prior to time 1.0 and unbalanced thereafter;

FIG. 12 is similar to FIG. 11, albeit illustrating a DC bus voltagewaveform generated using a controller operated in a manner consistentwith the present invention;

FIG. 13 is a graph illustrating steady state phase voltage and currentwaveforms generated using a conventional dq current loop controllerwhere supply line voltages were balanced prior to time 1.0 andunbalanced thereafter; and

FIG. 14 is a graph similar to FIG. 13, albeit illustrating waveformsgenerated using a controller operated in a manner consistent with thepresent invention.

DETAILED DESCRIPTION OF THE INVENTION

One or more specific embodiments of the present invention will bedescribed below. It should be appreciated that in the development of anysuch actual implementation, as in any engineering or design project,numerous implementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness related constraints, which may vary from one implementation toanother. Moreover, it should be appreciated that such a developmenteffort might be complex and time consuming, but would nevertheless be aroutine undertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure.

Hereinafter, subscript “a”, “b”, and “c” will be used to refer to supplyline voltage and current values in a three phase stationary frame ofreference, subscript, “u”, “v” and “w” will be used to refer tocontroller voltage and current values in a three phase stationary frameof reference. In addition, a subscript “f” is used to refer to afeedback current or voltage value, a subscript “ff” is used to refer toa feedforward value, a subscript “ref” is used to refer to a referencevalue, a superscript “*” is used to refer to a command value, asubscript “RMS” is used to refer to an RMS value, a subscript “L” isused to refer to a load value, a subscript “DC” is used to refer to a DCvalue, subscript “d” and “q” are used to refer to d and q-axis values ina two phase synchronous reference frame, respectively, subscript “α” and“β” are used to refer to α and β-axis values in a two phase stationaryreference frame and a subscript “o” is used to refer to an offset value.

Hereinafter, first some theory that forms the basis for the presentinventive concepts is provided, second, some practical controllerconfigurations are described that may be used to facilitate theinventive concepts and finally some graphs are described that illustratethe effects of the invention.

Referring now to FIG. 1, a simplified active converter configuration 8is illustrated and includes a three phase AC power source 10 thatprovides three phase voltages V_(a), V_(b) and V_(c) on three powerlines 12, 14 and 16 to a converter 15 including six power switchingdevices (e.g., semiconductor switching devices) 18, 12, 22, 24, 26 and28 which converts the three phase AC voltages to DC voltage V_(DC)across positive and negative DC buses 36 and 38, respectively. Each line12, 14 and 16 is characterized by a line inductance L and a lineresistance R.

Converter switches 18, 20, 22, 24, 26 and 28 are linked between thepositive and negative DC buses 36 and 38, respectively. A DC buscapacitor 40 is also linked between DC buses 36 and 38. As well known inthe conversion arts, converter switches 18, 20, 22, 24, 26 and 28 arecontrolled by a controller (not illustrated in FIG. 1) in a fashion togenerally convert the AC power on lines 12, 14 and 16 to DC across buses36 and 38. When switches 22-28 are controlled to generate DC voltageV_(DC), three phase voltages V_(u), V_(v) and V_(w) result at nodes 30,32 and 34, respectively. The potential differences across lines 12, 14and 16 (i.e., across inductive values L and resistive values R) causecurrents I_(u), I_(v) and I_(w) to pass threrethrough. Currents I_(u),I_(v) and I_(w) cause two phase q-axis current I_(q) which chargescapacitor 40. Although not illustrated in FIG. 1, a load (e.g., aninverter and motor/generator may be linked to buses 36 and 38 to receivepower therefrom where the load draws current I_(L).

The voltages V_(a), V_(b) and V_(c) in FIG. 1 can be represented by thefollowing equations:V _(a) =a·Sin (ωt)  Eq. 1V _(b) =b·Sin (ωt−120°)  Eq. 2V _(c) =c·Sin (ωt+120°)  Eq. 3where a, b and c are phase voltage peak values. If voltage source 10 isbalanced then a=b=c=e₀ where e₀ is the phase voltage peak for a balancedAC line voltage source. Combining the above equations and usingexpressions consistent with the labels in FIG. 1, the followingexpressions can be formulated:

$\begin{matrix}{{V_{u} - V_{a}} = {{I_{u}{R \cdot \left( {1 + {Tp}} \right)}\mspace{14mu}{or}\mspace{14mu} I_{u}} = {\left( {V_{u} - V_{a}} \right) \cdot \frac{1}{R} \cdot \frac{1}{1 + {Tp}}}}} & {{Eq}.\mspace{14mu} 4} \\{{V_{v} - V_{b}} = {{I_{v}{R \cdot \left( {1 + {Tp}} \right)}\mspace{14mu}{or}\mspace{14mu} I_{v}} = {\left( {V_{v} - V_{b}} \right) \cdot \frac{1}{R} \cdot \frac{1}{1 + {Tp}}}}} & {{Eq}.\mspace{14mu} 5} \\{{V_{w} - V_{c}} = {{I_{u}{R \cdot \left( {1 + {Tp}} \right)}\mspace{14mu}{or}\mspace{14mu} I_{w}} = {\left( {V_{w} - V_{c}} \right) \cdot \frac{1}{R} \cdot \frac{1}{1 + {Tp}}}}} & {{Eq}.\mspace{14mu} 6} \\{V_{DC} = {{\frac{1}{C} \cdot {\int{{\left( {I_{q} - I_{L}} \right) \cdot \ {\mathbb{d}t}}\mspace{14mu}{or}\mspace{14mu} V_{DC}}}} = {\left( {I_{q} - I_{L}} \right) \cdot \frac{1}{pC}}}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$where p is a differential operator (i.e., p=d/dt), T and is a timeconstant associated with values L and R (i.e., T=L/R), I_(q) is the twophase synchronous q-axis component output of an active converter currentand I_(L) is the load current. While the circuit of FIG. 1 illustrates athree phase system, many controllers operate in the synchronous twophase dq reference frame. For this reason, according to at least someinventive embodiments, signals and values are converted to the two phasedq reference frame. To this end, conversion can be performed via a twostep process. First, the three phase stationary values can be convertedto two phase stationary values in an αβ reference frame and then the twophase stationary values can be converted to the two phase synchronousvalues in the dq reference frame.

Equations 4, 5 and 6 can be converted from the three phase stationaryreference frame to the two phase synchronous dq reference frame usingthe following equations:

$\begin{matrix}{\alpha = a} & {{Eq}.\mspace{14mu} 8} \\{\beta = {{\frac{1}{\sqrt{3}}a} + {\frac{2}{\sqrt{3}}c}}} & {{Eq}.\mspace{14mu} 9} \\{d = {{\alpha\mspace{14mu} A\mspace{14mu}{{Cos}\left( {\omega\; t} \right)}} - {\beta\mspace{14mu} A\mspace{14mu}{{Sin}\left( {\omega\; t} \right)}}}} & {{Eq}.\mspace{14mu} 10} \\{q = {{\alpha\mspace{14mu} A\mspace{14mu}{{Sin}\left( {\omega\; t} \right)}} + {\beta\mspace{14mu} A\mspace{14mu}{{Cos}\left( {\omega\; t} \right)}}}} & {{Eq}.\mspace{14mu} 11}\end{matrix}$Combining Equations 1-6, 8 and 9 and manipulating the results yields thefollowing equations in the two phase αβ reference frame:V _(α) =I _(α) R A (1+Tp)+e ₀ A Sin (ωt)  Eq. 12V _(β) =I _(β) R A (1+Tp)+e ₀ A Cos (ωt)  Eq. 13Equations 10 through 13 can be combined to yield the followingequations:

$\begin{matrix}{V_{d} = {{I_{d}{R \cdot \left( {1 + {Tp}} \right)}\mspace{14mu}{or}\mspace{14mu} I_{d}} = {V_{d} \cdot \frac{1}{R} \cdot \frac{1}{1 + {Tp}}}}} & {{Eq}.\mspace{14mu} 14} \\{V_{q} = {{{I_{q}{R \cdot \left( {1 + {Tp}} \right)}} + {e_{0}\mspace{14mu}{or}\mspace{14mu} I_{q}}} = {\left( {V_{q} - e_{0}} \right) \cdot \frac{1}{R} \cdot \frac{1}{1 + {Tp}}}}} & {{Eq}.\mspace{14mu} 15}\end{matrix}$

Consistent with Equations 14 and 15 above, controllers that operate inthe dq reference frame typically include a d-component feedback currentloop, a q-component feedback current loop and a feed forward loop withinthe q-current loop that adds the peak of phase voltage e₀ within theloop. If voltage source 10 is balanced, all three currents I_(u), I_(v)and I_(w) are identical and the DC bus voltage has nominal ripple.

If the AC line voltage source has unbalanced voltages, currents I_(u),I_(v) and I_(w) will not be identical and therefore the converter cannotbe used at full capacity. If the line voltages are unbalanced, then peakvalues a, b, c and e₀ are not equal. To determine how to compensate forthe unbalanced line voltages, the magnitude of the unbalance must beidentified.

Referring to FIG. 2, a phase diagram 9 is illustrated that includes athree phase stationary reference frame including axis a, b and c and atwo phase stationary reference frame including axis α and β. Thefollowing equations can be written that are consistent with therelationships between the phasors illustrated in FIG. 2:

$\begin{matrix}{V_{\alpha} = {\frac{2}{3}\left( {V_{a} + V_{\alpha\; b} + V_{\alpha\; c}} \right)}} & {{Eq}.\mspace{14mu} 16} \\{V_{\beta} = {\frac{2}{3}\left( {V_{\beta\; b} - V_{\beta\; c}} \right)}} & {{Eq}.\mspace{14mu} 17} \\{V_{\alpha\; b} = {{{- V_{c}}{Cos}\; 60{^\circ}} = {{- \frac{1}{2}}V_{b}}}} & {{Eq}.\mspace{14mu} 18} \\{V_{\alpha\; c} = {{{- V_{c}}{Cos}\; 60{^\circ}} = {{- \frac{1}{2}}V_{c}}}} & {{Eq}.\mspace{14mu} 19} \\{V_{\beta\; b} = {{{- V_{b}}{Cos}\; 30{^\circ}} = {{- \frac{\sqrt{3}}{2}}V_{b}}}} & {{Eq}.\mspace{14mu} 20} \\{V_{\beta\; c} = {{{- V_{c}}{Cos}\; 30{^\circ}} = {{- \frac{\sqrt{3}}{2}}V_{c}}}} & {{Eq}.\mspace{14mu} 21}\end{matrix}$whereCombining Equations 16, 18 and 19 we can write the following equation:

$\begin{matrix}{V_{a} = {{e_{0}^{*} \cdot \left\lbrack \quad \right.}\left. \quad{{\frac{2}{3} \cdot a^{*} \cdot {{Sin}\left( {\omega\; t} \right)}} - \mspace{185mu}{\frac{1}{3} \cdot b^{*} \cdot {{Sin}\left( {{\omega\; t} - {120{^\circ}}} \right)}} - {\frac{1}{3} \cdot c^{*} \cdot {{Sin}\left( {{\omega\; t} + {120{^\circ}}} \right)}}} \right\rbrack}} & {{{Eq}.\mspace{11mu} 22}\;}\end{matrix}$where:

$e_{0}^{*} = \frac{a + b + c}{3}$and is the average peak of phase voltage and Eq. 23

${a^{*} = \frac{a}{e_{0}^{*}}};{b^{*} = \frac{b}{e_{0}^{*}}};{c^{*} = \frac{c}{e_{0}^{*}}}$where a*, b* and c* are the per unit peaks of phase voltages. Equation22 can be rewritten and manipulated to yield:

$\begin{matrix}{V_{\alpha} = {e_{0}^{*} \cdot \left\lbrack {{\frac{1}{6} \cdot \left( {{4\; a^{*}} + b^{*} + c^{*}} \right) \cdot {{Sin}\left( {\omega\; t} \right)}} + {\frac{1}{2\sqrt{3}} \cdot \left( {b^{*} - c^{*}} \right) \cdot {{Cos}\left( {\omega\; t} \right)}}} \right\rbrack}} & {{Eq}.\mspace{14mu} 24}\end{matrix}$If a*=b*=c*=1 for a balanced system then V_(α)=e₀ A Sin (ωt). Ifa*≠b*≠c* for an unbalanced system then Equation 24 can be expressed as:

$\begin{matrix}{{V_{\alpha} = {A_{\alpha}A\;{{Sin}\left( {{\omega\; t} + \varphi_{\alpha}} \right)}}}{{where}\text{:}}} & {{Eq}.\mspace{14mu} 25} \\{\begin{matrix}{A_{\alpha} = {e_{0}^{*} \cdot \sqrt{\left\lbrack {\frac{1}{6} \cdot \left( {{4\; a^{*}} + b^{*} + c^{*}} \right)} \right\rbrack^{2} + \left\lbrack \frac{b^{*} - c^{*}}{2\sqrt{3}} \right\rbrack^{2}}}} \\{= {\frac{e_{0}^{*}}{3} \cdot \sqrt{\left( {a^{*} + b^{*}} \right)^{2} + \left( {a^{*} + c^{*}} \right)^{2} + {2\left( a^{*} \right)^{2}} - {b^{*} \cdot c^{*}}}}}\end{matrix}{{and}\text{:}}} & {{Eq}.\mspace{14mu} 26} \\{\varphi_{\alpha} = {{{arc}\;{{tg}\left\lbrack \frac{6 \cdot \left( {b^{*} - c^{*}} \right)}{2{\sqrt{3} \cdot \left( {{4\; a^{*}} + b^{*} + c^{*}} \right)}} \right\rbrack}} = {{arc}\;{{tg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} - c^{*}} \right)}{{4\; a^{*}} + b^{*} + c^{*}} \right\rbrack}}}} & {{Eq}.\mspace{14mu} 27}\end{matrix}$

In a similar fashion, Equations 17, 20 and 21 can be combined to yieldthe following expression:

$\begin{matrix}{V_{\beta} = {e_{0}^{*} \cdot \begin{bmatrix}{{{\frac{1}{\sqrt{3}} \cdot c^{*}}{{Sin}\left( {\omega\; t} \right)}{Cos}\; 120{^\circ}} + {{\frac{1}{\sqrt{3}} \cdot c^{*}}{{Cos}\left( {\omega\; t} \right)}{Sin}\; 120{^\circ}} -} \\{{{\frac{1}{\sqrt{3}} \cdot b^{*}}{{Sin}\left( {\omega\; t} \right)}{Cos}\; 120{^\circ}} + {{\frac{1}{\sqrt{3}} \cdot b^{*}}{{Cos}\left( {\omega\; t} \right)}{Sin}\; 120{^\circ}}}\end{bmatrix}}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$which can be manipulated to yield:

$\begin{matrix}{V_{\beta} = {e_{0}^{*} \cdot \left\lbrack {{{\frac{\left( {b^{*} - c^{*}} \right)}{2\sqrt{3}} \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)} + {{\frac{1}{2} \cdot \left( {b^{*} + c^{*}} \right) \cdot {Cos}}\mspace{11mu}\left( {\omega\; t} \right)}} \right\rbrack}} & {{Eq}.\mspace{14mu} 29}\end{matrix}$If a*=b*=c*=1 for a balanced system, then V_(β)=e₀ A Cos (ωt). Ifa*≠b*≠c* for an unbalanced system then Equation 29 can be expressed as:V _(β) =A _(β) A Sin (ωt+φ _(β))  Eq. 30

$\begin{matrix}{{where}:} & \; \\{\begin{matrix}{A_{\beta} = {e_{0}^{*} \cdot \sqrt{\left\lbrack \frac{b^{*} - c^{*}}{2\sqrt{3}} \right\rbrack^{2} + \left\lbrack \frac{b^{*} + c^{*}}{4} \right\rbrack^{2}}}} \\{= {\frac{e_{0}^{*}}{\sqrt{3}} \cdot \sqrt{\left( {b^{*} + c^{*}} \right)^{2} - {b^{*} \cdot c^{*}}}}}\end{matrix}{and}} & {{Eq}.\mspace{20mu} 31} \\{\varphi_{\beta} = {{arctg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} + c^{*}} \right)}{b^{*} - c^{*}} \right\rbrack}} & {{Eq}.\mspace{20mu} 32}\end{matrix}$

Phase equations similar to Equations 12 and 13 for a system including abalanced source can be written for a system linked to an unbalancedsource as:V _(α) =I _(α) R·(1+Tp)+A _(α)·Sin (ωt+φ _(α))  Eq. 33V _(β) =I _(β) R·(1+Tp)+A _(β)·Sin (ωt+φ _(β))  Eq. 34

Equations 33 and 34 can be converted to the two phase synchronous dqframe of reference using the well known formulas:V _(d) =V _(α) Cos (ωt)−V _(β) Sin (ωt)  Eq. 35V _(q) =V _(α) Sin (ωt)+V _(β) Cos (ωt)  Eq. 36Equations 33 through 36 can be combined to yield:V _(d) =I _(d) R·(1+Tp)+A _(α)·Sin (ωt+φ _(α))·Cos (ωt)−A _(β)·Sin(ωt+φ_(β))·Sin(ωt)  Eq. 37V _(q) =I _(q) R·(1+Tp)+A _(α)·Sin (ωt+φ _(α))·Sin(ωt)+A _(β)·Sin(ωt+φ_(β))·Cos (ωt)  Eq. 38

In almost all cases it can be assumed that the unbalance between thesupply line phases will not be more than 25% as indicated by thefollowing expression:0.75<a*<1.25; 0.75<b*<1.25 and 0.75<c*<1.25  Eq. 39Here, if the assumption in Equation 39 is made, then:

$\begin{matrix}{{{A_{\alpha} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\alpha} \right)} \approx {{A_{\beta} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\beta} \right)}} & {{Eq}.\mspace{14mu} 40} \\{{{A_{\alpha} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\alpha} \right)} \approx {{A_{\beta} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\beta} \right)}} & {{Eq}.\mspace{14mu} 41} \\{{{{A_{\alpha} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\alpha} \right)} - {{A_{\beta} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\beta} \right)}} \approx 0} & {{Eq}.\mspace{14mu} 42} \\{{{{A_{\alpha} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\alpha} \right)} \approx {{A_{\beta} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\beta} \right)}} = {\frac{{{A_{\alpha} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\alpha} \right)} + {{A_{\beta} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\beta} \right)}}{2} = {A_{\max} \approx e_{0}^{*}}}} & {{Eq}.\mspace{14mu} 43} \\{{{{A_{\alpha} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\alpha} \right)} \approx {{A_{\beta} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\beta} \right)}} = {\frac{{{A_{\alpha} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\alpha} \right)} + {{A_{\beta} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\beta} \right)}}{2} = A_{\min}}} & {{Eq}.\mspace{14mu} 44} \\{{{{A_{\alpha} \cdot {Sin}}\mspace{11mu}\left( \varphi_{\alpha} \right)} + {{A_{\beta} \cdot {Cos}}\mspace{11mu}\left( \varphi_{\beta} \right)}} = {2A_{\min}}} & {{Eq}.\mspace{14mu} 45}\end{matrix}$Combining Equations 37, 38 and 40-45 yields the following equations:

$\begin{matrix}{I_{d} = {\left\lbrack {V_{d} - {{A_{\min} \cdot {Cos}}\mspace{11mu}\left( {2\omega\; t} \right)}} \right\rbrack \cdot \frac{1}{R \cdot \left( {1 + {Tp}} \right)}}} & {{Eq}.\mspace{14mu} 46} \\{I_{q} = {\left\{ {V_{q} - \left\lbrack {e_{0}^{*} + {{A_{\min} \cdot {Sin}}\mspace{11mu}\left( {2\omega\; t} \right)}} \right\rbrack} \right\} \cdot \frac{1}{R \cdot \left( {1 + {Tp}} \right)}}} & {{Eq}.\mspace{14mu} 47}\end{matrix}$

Equations 46 and 47 correspond to a d-q model or object corresponding toan active converter linked to an unbalanced AC line voltage source.Equations 46 and 47 show that both d and q active converter currentsinclude a second harmonic component and therefore a control scheme thatuses phase peak e₀ as a simple feed forward control is insufficient toeliminate the second harmonics in the supply line voltages that occurduring active conversion of unbalanced voltages. Consistent withEquations 46 and 47, a more complicated feed forward compensation schemefor each of the d and q-axis loops is required.

Calculation of the feed forward signal values can be made based onequations 26, 27, 31, 32 and 44. These equations require knowledge aboutthe phase peak voltage (i.e., a, b and c) for each individual phase ofthe unbalanced AC line voltage source. Typical control configurationsinclude sensors that measure RMS line-to-line voltages as opposed topeak phase voltages. To facilitate compensation with existingconventional hardware, phase AC voltages must be derived from RMSline-to-line voltages (i.e., peak values a, b and c have to be expressedas a function of RMS line-to-line values V_(ab) _(—) _(RMS), V_(bc) _(—)_(RMS), V_(ca) _(—) _(RMS)).

Combining Equations 1-3, line-to-line voltages V_(ab), V_(bc), andV_(ca) can be expressed as:

$\begin{matrix}\begin{matrix}{V_{ab} = {V_{a} - V_{b}}} \\{= {{{a \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)} - {{b \cdot {Sin}}\mspace{11mu}\left( {{\omega\; t} - 120^{{^\circ}}} \right)}}} \\{= {{{a \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)} - \left\lbrack {{{b \cdot {Sin}}\mspace{11mu}{\left( {\omega\; t} \right) \cdot {Cos}}\mspace{11mu}\left( 120^{{^\circ}} \right)} -} \right.}} \\\left. {{b \cdot {Cos}}\mspace{11mu}{\left( {\omega\; t} \right) \cdot {Sin}}\mspace{11mu}\left( 120^{{^\circ}} \right)} \right\rbrack \\{= {{{\left( {a + \frac{b}{2}} \right) \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)} + {{\frac{\sqrt{3}}{2} \cdot b \cdot {Cos}}\mspace{11mu}\left( {\omega\; t} \right)}}} \\{= {\sqrt{a^{2} + {a \cdot b} + b^{2}} \cdot {{Sin}\left\lbrack {{\omega\; t} + {{arctg}\mspace{11mu}\left( \frac{\sqrt{3} \cdot b}{{2 \cdot a} + b} \right)}} \right\rbrack}}}\end{matrix} & {{Eq}.\mspace{14mu} 48} \\\begin{matrix}{V_{bc} = {V_{b} - V_{c}}} \\{= {{{{b \cdot {Sin}}\mspace{11mu}\left( {{\omega\; t} - 120^{{^\circ}}} \right)} - {{c \cdot {Sin}}\mspace{11mu}\left( {{\omega\; t} + 120^{{^\circ}}} \right)}} =}} \\{= {{{{- \frac{1}{2}} \cdot \left( {b - c} \right) \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)} - {{\frac{\sqrt{3}}{2} \cdot \left( {b + c} \right) \cdot {Cos}}\mspace{11mu}\left( {\omega\; t} \right)}}} \\{= {\sqrt{b^{2} + {b \cdot c} + c^{2}} \cdot {{Sin}\left\lbrack {{\omega\; t} + {{arctg}\mspace{11mu}\left( \frac{\sqrt{3} \cdot \left( {b + c} \right)}{b - c} \right)}} \right\rbrack}}}\end{matrix} & {{Eq}.\mspace{14mu} 49} \\\begin{matrix}{V_{ca} = {V_{c} - V_{a}}} \\{= {{{c \cdot {Sin}}\mspace{11mu}\left( {{\omega\; t} + 120^{{^\circ}}} \right)} - {{a \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)}}} \\{= {{{c \cdot {Sin}}\mspace{11mu}{\left( {\omega\; t} \right) \cdot {Cos}}\mspace{11mu}\left( 120^{{^\circ}} \right)} +}} \\{{{c \cdot {Cos}}\mspace{11mu}{\left( {\omega\; t} \right) \cdot {Sin}}\mspace{11mu}\left( 120^{{^\circ}} \right)} - {{a \cdot {Sin}}\mspace{11mu}\left( {\omega\; t} \right)}} \\{= {{{- {\cdot \left( {a + \frac{c}{2}} \right) \cdot {Sin}}}\mspace{11mu}\left( {\omega\; t} \right)} + {{\frac{\sqrt{3}}{2} \cdot c \cdot {Cos}}\mspace{11mu}\left( {\omega\; t} \right)}}} \\{= {\sqrt{a^{2} + {a \cdot c} + c^{2}} \cdot {{Sin}\left\lbrack {{\omega\; t} + {{arctg}\mspace{11mu}\left( \frac{\sqrt{3} \cdot c}{{2 \cdot a} + c} \right)}} \right\rbrack}}}\end{matrix} & {{Eq}.\mspace{14mu} 50}\end{matrix}$respectively. If the AC line voltage source is balanced then the peakvalues will be equal (i.e., a=b=c) and V_(ab)=√{square root over(3)}·a·Sin (ωt+30°); V_(bc)=√{square root over (3)}·a·Sin (ωt+90°) andV_(ca)=−√{square root over (3)}·a·Sin (ωt−30°). The RMS line-to-linevoltages V_(ab) _(—) _(RMS), V_(bc) _(—) _(RMS, V) _(ca) _(—) _(RMS) foran unbalanced system can be expressed as:

$\begin{matrix}{V_{ab\_ RMS} = \frac{\sqrt{a^{2} + {a \cdot b} + b^{2}}}{\sqrt{2}}} & {{Eq}.\mspace{14mu} 51} \\{V_{bc\_ RMS} = \frac{\sqrt{b^{2} + {b \cdot c} + c^{2}}}{\sqrt{2}}} & {{Eq}.\mspace{14mu} 52} \\{V_{ca\_ RMS} = \frac{\sqrt{a^{2} + {a \cdot c} + c^{2}}}{\sqrt{2}}} & {{Eq}.\mspace{14mu} 53}\end{matrix}$respectively. Equations 51-53 can be rewritten as:a ² +a·b+b ²=2·V ² _(ab) _(—) _(RMS)  Eq. 54b ² +b·c+c ²=2·V ² _(bc) _(—) _(RMS)  Eq. 55c ² +c·a+a ²=2·V ² _(ca) _(—) _(RMS)  Eq. 56

A general equation for the relationship between phase peak values a, band c and line-to-line RMS voltages with either a balanced or anunbalanced voltage source can be expressed as:

$\begin{matrix}{{a + b + c} = \frac{\sqrt{2} \cdot \left( {V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}} \right)}{\sqrt{3}}} & {{Eq}.\mspace{14mu} 57}\end{matrix}$Subtracting Equation 55 from Equation 56 and taking Equation 57 intoaccount yields the following equation:

$\begin{matrix}{{a - b} = {\frac{\sqrt{6} \cdot \left( {V_{ca\_ RMS}^{2} - V_{bc\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}} = X_{1}}} & {{Eq}.\mspace{14mu} 58}\end{matrix}$Similarly, Subtracting Equation 56 from Equation 54 and taking Equation57 into account yields the following equation:

$\begin{matrix}{{b - c} = {\frac{\sqrt{6} \cdot \left( {V_{ab\_ RMS}^{2} - V_{ca\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}} = X_{2}}} & {{Eq}.\mspace{14mu} 59}\end{matrix}$and, subtracting Equation 54 from Equation 55 and taking Equation 57into account yields the following equation:

$\begin{matrix}{{c - a} = {\frac{\sqrt{6} \cdot \left( {V_{bc\_ RMS}^{2} - V_{ab\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}} = X_{3}}} & {{Eq}.\mspace{14mu} 60}\end{matrix}$Equations 54 and 58 can be solved together. To this end, from Equation58:b=a−X ₁  Eq. 61Combining Equations 61 and 54 and rearranging the terms:

$\begin{matrix}{a = {\frac{X_{1}}{2} + \sqrt{{{\frac{2}{3} \cdot V^{2}}{ab}_{-}{RMS}} - \frac{X_{1}^{2}}{12}}}} & {{Eq}.\mspace{14mu} 62}\end{matrix}$If we will take into account that

${\frac{X_{1}^{2}}{12}{{\operatorname{<<}\frac{2}{3}} \cdot V^{2}}{ab}_{-}{RMS}},$then Equation 62 can be simplified as:

$\begin{matrix}{a \approx {\frac{X_{1}}{2} + {\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}}}} & {{Eq}.\mspace{14mu} 63}\end{matrix}$Equations 58 and 63 can be combined to identify the peak phase value aas:

$\begin{matrix}{a = {{\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ca}_{-}{RMS}}^{2} + V_{{bc}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}} & {{Eq}.\mspace{14mu} 64}\end{matrix}$

The same substitutions and manipulations as above can be made for eachof the other two phases to identify peak values b and c according to thefollowing equations:

$\begin{matrix}{b = {{\sqrt{\frac{2}{3}} \cdot V_{{bc}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ab}_{-}{RMS}}^{2} - V_{{ca}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}} & {{Eq}.\mspace{14mu} 65} \\{c = {{\sqrt{\frac{2}{3}} \cdot V_{{ca}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{bc}_{-}{RMS}}^{2} - V_{{ab}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{cz}_{-}{RMS}}}}}} & {{Eq}.\mspace{14mu} 66}\end{matrix}$

First Embodiment

Referring now to FIG. 3, an exemplary controller 50 linked to aconverter 15 is illustrated. Controller 50 receives a DC bus voltagereference value V_(DCref) and controls converter 15 so that converter 15generates a DC bus voltage V_(DC) (see again FIG. 1) that is equal tothe reference voltage V_(DCref). In this regard, as illustrated in FIGS.1 and 3, three phase supply line voltages V_(a), V_(b) and V_(c) areprovided to converter 15 and converter 15 is controlled in a mannerconsistent with command voltage values V*_(u), V*_(v) l and V*_(w) tocause the DC bus voltage to track the reference voltage V_(DCref).

In addition to the controller 50 and converter 15, a synchronous twophase dq reference frame object model 52 is illustrated for discussionpurposes to simulate converter operation when unbalanced supply linevoltages occur on lines 12, 14 and 16. Object model 52 includes an angleθ determiner 60, two stationary three-phase to synchronous two phasetransformers 56 and 58, three summers 62, 64 and 66 and three scalarmodules 68, 70 and 71.

As its label implies, angle determiner 60 is linked to the three phasesupply lines 12, 14 and 16 and identifies a supply voltage angle θ whichis provided to each of transformers 56 and 58. Transformer 56 transformsthe stationary three-phase voltage values V_(u), V_(v) and V_(w) fromconverter 15 to two-phase synchronous voltages V_(q) and V_(d) in the dqreference frame. Similarly, transformer 58 transforms the supply linevoltage values V_(a), V_(b) and V_(c) into d and q values that includesecond harmonic components. Here, as illustrated and, consistent withthe discussion above, the d and q second harmonic components arerepresented by values A_(min) Cos (2ωt) and A_(min) Sin (2ωt),respectively. In addition to the second harmonic component A_(min) Sin(2ωt), the q-axis value includes a DC offset e₀.

Summer 62 subtracts value e₀+A_(min) Sin (2ωt) from q-axis voltage V_(q)thereby generating a q-axis difference voltage value ΔV_(q). Similarly,summer 64 subtracts d-axis second harmonic value A_(min) Cos (2ωt) fromd-axis voltage V_(d) thereby generating a d-axis difference voltagevalue ΔV_(d). The difference values ΔV_(q) and ΔV_(d) are provided tomodules 68 and 70, respectively, where each is multiplied by a scalarconsistent with Equations 46 and 47 described above thereby generating qand d-axis currents l_(q) and l_(d), respectively. Q-axis current l_(q)is provided to third summer 66 which subtracts a load current l_(L) andprovides it's output to module 71. Module 71 divides the value receivedfrom summer 66 by the derivative of the value of capacitor 40 (see againFIG. 1) thereby providing a DC bus voltage value V_(DC).

Referring still to FIG. 3, each of the d and q-axis currents I_(d) andI_(q) are provided as feedback currents I_(df) and I_(qf) to controller50. While this feedback process is shown in a simplified form in FIG. 3,it should be appreciated that in an actual implementation the feedbackprocess would be more complicated and require measurement of supply linecurrents and conversion to the dq reference frame. In addition to the dand q currents, the bus voltage value V_(DC) is fed back to controller50 which includes a gain block 96 that steps up the bus value V_(DC) bya scalar K_(v) to a value V_(DCf) suitable for comparison to referencevalue V_(DCref).

Referring still to FIG. 3, controller 50 includes five summers 80, 82,84, 86 and 88, three proportional/integral (PI) regulators 90, 92 and94, a two phase synchronous to three phase stationary transformer 98, aline-to-line RMS measurement module 100, a feedforward voltagecalculator 102, a scalar module 104 and an integrator 106.

RMS measurement module 100 is linked to supply lines 12, 14 and 16 andmeasures the line-to-line RMS voltage values V_(ab) _(—) _(RMS), V_(bc)_(—) _(RMS) and V_(ca) _(—) _(RMS). In addition, RMS measurement module100 identifies the frequency of the supply voltages V_(a), V_(b) andV_(c) in hertz and provides that value to scalar module 104. The RMSvoltage values are provided to calculator 102. Calculator 102 uses theRMS voltage values to identify the DC offset e₀ and the d and q-axissecond harmonic components A_(min) cos(2ωt) and A_(min) sin(2ωt),respectively. The sum of the DC offset value and the q-axis secondharmonic component are provided to summer 84. Similarly, the d-axissecond harmonic component is provided to summer 88. Scalar module 104multiplies the frequency by 2π and provides its output to integrator106. Integrator 106 integrates the value received from module 104 andprovides an angle θ value to transformer 98.

Referring still to FIG. 3, summer 80 subtracts the DC feedback voltagevalue V_(DCf) from the DC reference voltage value V_(DCref) and providesits output to Pl regulator 90. Regulator 90 steps up the value receivedfrom summer 80 and provides its output as a q-axis command currentl*_(q) to summer 82. Summer 82 subtracts the q-axis feedback currentl_(qf) from the q-axis command current l*_(q) and provides its output toP regulator 92. Regulator 92 steps up the value it receives from summer82 and provides its output as a q-axis difference voltage value ΔV_(q)to summer 84. Difference value ΔV_(q) is akin to the output of summer 62in object 52 and therefore, if value e₀+A_(min) Sin (2ωt) is added tovalue ΔV_(q), the modification will compensate for the subsequent effectof summer 64 in object model 52. To this end, summer 84 adds valueΔV_(q) to the q-axis value (i.e., e₀+A_(min) Sin (2ωt)) received fromcalculator 102 and provides its output as a q-axis command voltageV*_(q) to transformer 98.

The d-axis feedback current l_(df) is subtracted from a d-axis commandcurrent l*_(d) by summer 86 which provides its output to Pl regulator94. Regulator 94 steps up the value received from summer 86 and providesits output as a d-axis difference voltage value ΔV_(d) to summer 88.Difference value ΔV_(d) is akin to the output of summer 64 in object 52and therefore, if value A_(min) Cos(2ωt) is added to value ΔV_(d), themodification will compensate for the subsequent effect of summer 62 inobject model 52. To this end, summer 88 adds the d-axis differencevoltage value ΔV_(d) to the d-axis second harmonic component A_(min)cos(2ωt) received from calculator 102 and provides its output as ad-axis command voltage value V*_(d) to transformer 98.

Transformer 98 transforms the q and d-axis voltage command values V*_(q)and V*_(d) to three-phase stationary command voltage values V*_(u),V*_(v) and V*_(w) which are provided to converter 54. Although notillustrated, converter 15 uses values V*_(u), V*_(v) and V*_(w) tocontrol switches (e.g., see 18, 20, 22, 24, 26 and 28) to generate theDC bus voltage V_(DC) across buses 36 and 38.

Referring to FIG. 4, exemplary V_(ff) calculator 102 is illustrated.Exemplary calculator 102 includes two scalar modules 120 and 122, anintegrator 124, a cosine module 126, a sine module 128, first and secondmultipliers 130 and 132, a summer 134 and five calculator modules 136,138, 140, 142 and 144. The supply voltage frequency is doubled by module120 which provides its output to module 122. Module 122 multiplies 2π bythe value received from module 120 and provides its output as the secondharmonic frequency to integrator 124. Integrator 124, as its labelimplies, integrates the value it receives and provides an angle valuecorresponding to the second harmonic of the supply voltages to each ofthe cosine and sine modules 126 and 128. Cosine module 126 generates thecosine of the received angle and provides that value to multiplier 130.Similarly, sine module 128 generates the sine of the angle received andprovides its output to multiplier 132.

Referring still to FIG. 4, first calculator module 136 receives each ofthe line-to-line RMS voltage values and solves equation 66 to identifypeak phase value c which is provided to fifth calculator module 144.Similarly, modules 138 and 140 receive the line-to-line RMS voltagevalues and solve equations 65 and 64, respectively, to identify the peakphase values b and a, respectively, each of which are provided to fifthcalculator module 144.

Calculator module 142 receives the line-to-line RMS voltage values andsolves equation 57 to identify the combined value of peak values a, band c. Thereafter, module 142 divides the combined value by three toidentify value e₀ in a manner consistent with equation 23 above.

Calculator module 144 solves equations 26, 27, 31, 32 and 44 to identifyamplitude value A_(min). Value A_(min) is provided to each ofmultipliers 130 and 132 and is multiplied by the cosine and sine valuesgenerated by modules 126 and 128, respectively. The output of multiplier130 is the d-axis second harmonic component A_(min) cos(2ωt). The outputof multiplier 132 is provided to summer 134 and is added to values e₀thereby generating the q-axis component e₀+A_(min) sin(2ωt) whichincludes the q-axis second harmonic component. Values A_(min) cos(2ωt)and e₀+A_(min) sin(2ωt) are provided to summers 88 and 84 as describedabove to pre-compensate for second harmonics that would occur on thesupply lines during active converter control when the supply linevoltages are unbalanced.

Second Embodiment

While most control configurations operate in the synchronous two-phasedq reference frame, it is possible to configure a controller thatoperates in the three phase stationary frame of reference. In this case,Equations 1 through 3 and Equations 64 through 66 can be used toidentify three phase feed forward voltages to substantially eliminatethe second harmonic components in the line voltages and to reduce DC busripple.

Referring now to FIG. 5, a schematic diagram similar to the diagramprovided in FIG. 3 as illustrated that includes a three phase objectmodel 152 to illustrate the effects of active conversion with unbalancedsupply line voltages, a three phase controller 150 and a converter 169.Converter 169 receives the supply line voltages V_(a), V_(b) and V_(c)and generates voltage values V_(u), V_(v) and V_(w) as a function ofcommand voltages V*_(u), V*_(v) and V*_(w).

Object model 152 includes four summers 157, 159, 161 and 166, an angledeterminer 160, four scalar modules 168, 170, 172 and 171 and a threephase stationary to two phase synchronous transformer 156. Summer 157subtracts the line voltage value V_(a) from node voltage value V_(u) andprovides its output to module 168. Similarly, summer 159 subtracts theline voltage value V_(b) from node voltage value V_(v) and provides itsoutput to module 170 while summer 161 subtracts line voltage value V_(c)from node voltage value V_(w) and provides its output to module 172. Theoutputs of summers 157, 159 and 161 are three phase difference voltagevalues ΔV_(a), ΔV_(b) and ΔV_(c), respectively. Referring once again toFIG. 1, the difference values ΔV_(a), ΔV_(b) and ΔV_(c) represent thepotentials across the inductive and resistive values in line 12, 14 and16. Each of modules 168, 170 and 172 multiplies the received differencevalue by a scalar consistent with Equations 4, 5 and 6 described abovethereby generating values corresponding to three phase line currentsI_(u), I_(v) and I_(w). Current values I_(u), I_(v) and I_(w) areprovided to transformer 156 which transforms those three phasestationary values to two phase synchronous values I_(q) and I_(d). InFIG. 5, value I_(d) is not shown as that value is not important from theperspective of object 152 and how that object is employed for thepurposes of the present invention. Q-axis current value I_(q) isprovided to summer 166. Summer 166 subtracts the load current l_(L) fromthe q-axis current I_(q) and provides its output to scalar module 171.Module 171 divides the value received from summer 166 by the derivativeof the value of capacitor 40 (see again FIG. 1) thereby providing a DCbus voltage value V_(DC).

Referring still to FIG. 5, each of the three phase line current valuesl_(u), l_(v) and l_(w) are stepped up by a gain module 173, the steppedup values being provided as three phase feedback currents l_(vf), l_(uf)and l_(wf) to controller 150. In addition, the bus voltage value V_(DC)is fed back to controller 150 that includes a gain block 196 that stepsup the bus value V_(DC) by a scalar K_(v) to a value V_(DCf) suitablefor comparison to reference value V_(Dcref).

Controller 150 includes eight summers 180, 186, 182, 184, 188, 151, 153and 155, five Pl regulators 190, 191, 185, 187 and 189, two signaltransformers 177 and 193, a line-to-line RMS measurement module 200, afeed forward voltage calculator 202, a scalar module 204 and anintegrator 206.

RMS measurement module 200 is akin to module 100 in FIG. 3 and providesas outputs line-to-line RMS voltage values as well as a line frequencyvalue in hertz. Calculator 202 uses the RMS voltage values to identifythree phase feed forward voltage values V_(aff), V_(bff) and V_(cff)which are provided to summers 151, 153 and 155, respectively. Scalarmodule 204 multiplies the frequency received from module 200 by 2π andprovides its output to integrator 206. Integrator 206 integrates thevalue received from module 204 and provides an angle value totransformer 177.

Referring still to FIG. 5, summer 180 subtracts the DC feedback voltagevalue V_(DCf) from the DC reference voltage value V_(Dcref) and providesits output to regulator 190. Regulator 190 steps up the value receivedfrom summer 180 and provides its output as a q-axis command currentl*_(q) to transformer 177.

The feedback current values l_(vf), l_(uf) and l_(wf) are provided tothe three-to-two phase transformer 193 which generates a d-axis feedbackcurrent value l_(df). Summer 186 subtracts the d-axis feedback currentl_(df) from a d-axis reference current l_(dref) and provides thedifference to Pl regulator 191. Regulator 191 steps up the valuereceived from summer 186 and provides its output as a d-axis commandcurrent l*_(d) to transformer 177.

Transformer 177 transforms the d and q-axis command currents l*_(d) andl*_(q) to three phase stationary reference current values l_(uref),l_(vref) and l_(wref) which are provided to summers 188, 184 and 182.Summer 188 subtracts feedback current l_(uf) from reference currentl_(uref) and provides its output as a command current I*_(u) tocontroller 189. Similarly, summer 184 subtracts feedback current l_(vf)from reference current l_(vref) and provides its output as a commandcurrent I*_(v) to controller 187 while summer 182 subtracts feedbackcurrent l_(wf) from reference current l_(wref) and provides its outputas a command current I*_(w) to controller 185. Each of controllers 185,187 and 189 steps up the value received and provides a differencevoltage value as an output. To this end, controller 185 providesdifference value ΔV_(w), controller 187 provides difference value ΔV_(v)and controller 189 provides value ΔV_(u) to summers 155, 153, and 151respectively. Summer 151 adds the difference value ΔV_(u) and thefeedforward voltage value V_(aff) and provides its output as a commandvoltage value V*_(u) to converter 169. Similarly, summer 153 addsdifference value ΔV_(v) and feedforward voltage value V_(bff) providingits output is a command voltage value V*_(v) and summer 155 addsdifference value ΔV_(w) and feedforward voltage V_(cff) and provides itsoutput as command voltage value V*_(w) to converter 169.

Referring now to FIG. 6, exemplary calculator 202 is illustrated.Exemplary calculator 202 includes one scalar module 222, an integrator224, three sign modules 226, 227 and 228, three multipliers 230, 231 and232 and three calculator modules 236, 238 and 240. Module 222 multiplies2π by the supply line frequency and provides its output to integrator224 which, as its label implies, integrates the value received andprovides an angle value to each of sign modules 226, 227 and 228. Signmodule 228 generates the sign of the angle received and provides itsoutput to multiplier 232. Module 227 generates the sign of the anglereceived less 120 degrees and provides its output to multiplier 231.Module 226 generates the sign of the received angle plus 120 degrees andprovides its output to multiplier 230. At this point it should beappreciated that the outputs of modules 228, 227 and 226 provide aportion of the solution to equations 1, 2 and 3 above. Calculatormodules 240, 238 and 236 provide the other parts of equations 1-3 above.To this end, modules 240, 238 and 236 solve equations 64, 65 and 66 toidentify the peak voltage values a, b and c, respectively. Values a, band c are provided to multipliers 232, 231 and 230 where they aremultiplied by the outputs of modules 228, 227 and 226 to providefeedforward voltage values V_(aff), V_(bff) and V_(cff), respectively.Values V_(aff), V_(bff) and V_(cff) are provided to summers 151, 153 and155 to compensate for the unbalance in line voltages V_(a), V_(b) andV_(c).

Referring now to FIG. 7, u, v and w phase RMS currents are illustratedthat were generated using a conventional dq current loop controllerwhere the AC line voltage source was balanced up to time 1 andthereafter was unbalanced. In the illustrated example, phase u wasunbalanced by −10 percent, phase v was unbalanced by −8 percent andphase w was unbalanced by +4 percent. Clearly at time 1, the unbalancecauses unbalanced RMS currents to occur.

Referring to FIG. 8, three phase RMS currents similar to those of FIG. 7are illustrated where, again, the voltage source was balanced up to time1 and thereafter was unbalanced to a similar degree as that describedabove. In FIG. 8, however, a controller operated in a manner consistentwith the present invention described above was employed that controlledindividual phase currents in a three phase manner. Clearly the unbalanceshown in FIG. 7 is substantially reduced in FIG. 8.

Referring to FIG. 9, FIG. 9 illustrates three phase steady statecurrents when the line voltages are unbalanced with a conventional dqcurrent loop controller. As illustrated, the currents change amplitudeand phase due to the unbalanced source with the conventional dq currentloop controller. In contrast, in FIG. 10 it can be seen that the steadystate currents that result when a controller operated in a mannerconsistent with the teachings of the present invention is used do notchange amplitude or phase.

Referring to FIG. 11, a DC bus voltage waveform is illustrated where thevoltage source was balanced until time 1 and thereafter was unbalanced.It can be seen that after time 1, a large second harmonic rippleappears. In contrast, referring to FIG. 12 where an individual phasecurrent control scheme was employed as consistent with the presentinvention, when the voltage source becomes unbalanced at time 1, whilethere is a short transient ripple, the ripple is quickly eliminated.

Referring now to FIG. 13, waveforms representing steady state phasevoltage V and current l that result from a conventional dq current loopcontroller where the AC line voltage source is unbalanced areillustrated. It should be appreciated that the power factor is not unitywhen the voltage source is unbalanced. In FIG. 14, voltage and currentwaveforms similar to those in FIG. 13 are illustrated where the AC linevoltage source was balanced up to time 1 and was unbalanced thereafter.The system used to generate the waveforms of FIG. 14 employed a controlscheme consistent with the present invention. As can be seen, the powerfactor associated with the waveforms of FIG. 14 is unity both when theline voltage source is balanced and unbalanced when the inventiveconcepts are employed.

While the invention may be susceptible to various modifications andalternative forms, specific embodiments have been shown by way ofexample in the drawings and have been described in detail herein.However, it should be understood that the invention is not intended tobe limited to the particular forms disclosed. Thus, the invention is tocover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the invention as defined by the followingappended claims.

To apprise the public of the scope of this invention, the followingclaims are made:

1. A method for use with a controller and a converter where thecontroller actively controls the converter to convert three phasevoltages on three supply lines to a DC voltage across positive andnegative DC buses, the method comprising the steps of: identifying thepeak amplitudes of the three phase supply line voltages; using the peakamplitudes to identify a second harmonic component that would begenerated on the supply lines by the converter during normal operationdue to unbalance in the peak amplitudes; and altering control of theconverter as a function of the identified second harmonic.
 2. The methodof claim 1 wherein the controller generates command voltages to controlthe converter, the step of altering control of the converter includingmodifying the command voltages as a function of the identified secondharmonic.
 3. The method of claim 2 wherein the step of identifying thepeak amplitudes includes sensing the RMS line-to-line voltages and usingthe RMS line-to-line voltages to identify the peak amplitudes.
 4. Themethod of claim 3 wherein the step of identifying the peak amplitudesincludes the step of solving the following equations:$a = {{\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ca}_{-}{RMS}}^{2} - V_{{bc}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$$b = {{\sqrt{\frac{2}{3}} \cdot V_{{bc}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ab}_{-}{RMS}}^{2} - V_{{ca}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$${c = {{\sqrt{\frac{2}{3}} \cdot V_{{ca}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{bc}_{-}{RMS}}^{2} - V_{{ab}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}},$where a, b and c are the peak amplitudes and where V_(ab—RMS),V_(bC—RMS) and V_(ca—RMS)are the first, second and third RMSline-to-line voltages that occur between the first and second, secondand third and third and first lines, respectively.
 5. The method ofclaim 2 wherein the controller receives a DC reference voltage andcontrols the converter to cause the DC voltage across the DC buses totrack the reference voltage, the method further including the step ofusing the reference voltage to identify d and q-axis voltage differencevalues, the step of identifying the second harmonic components includingthe step of identifying d and q-axis components of the second harmonic,the step of modifying the command voltages including mathematicallycombining the difference values and the second harmonic components toidentify two phase d and q-axis command voltages.
 6. The method of claim5 wherein the step of mathematically combining includes adding the d andq-axis second harmonic components to the d and q-axis difference values.7. The method of claim 6 further including the step of using the peakamplitudes to identify a DC offset and wherein the step of modifying thecommand voltages further includes adding the DC offset to the q-axisdifference value along with the q-axis second harmonic component.
 8. Themethod of claim 5 wherein the peak amplitudes of the three supply linevoltages are a, b and c and wherein the step of identifying the d and qaxis second harmonic components includes using the peak amplitudes toidentify a two phase amplitude A_(min) by solving the followingequation:$A_{\min} = \frac{{A_{\alpha} \cdot {{Sin}\left( \varphi_{\alpha} \right)}} + {A_{\beta} \cdot {{Cos}\left( \varphi_{\beta} \right)}}}{2}$where:$A_{\alpha} = {\frac{e_{0}^{*}}{3} \cdot \sqrt{\left( {a^{*} + b^{*}} \right)^{2} + \left( {a^{*} + c^{*}} \right)^{2} + {2\left( a^{*} \right)^{2}} - {b^{*} \cdot c^{*}}}}$$A_{\beta} = {\frac{e_{0}^{*}}{\sqrt{3}} \cdot \sqrt{\left( {b^{*} + c^{*}} \right)^{2} - {b^{*} \cdot c^{*}}}}$$\underset{\alpha}{\varphi} = {{arc}\;{{tg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} - c^{*}} \right)}{{4a^{*}} + b^{*} + c^{*}} \right\rbrack}}$$\varphi_{\beta} = {{arc}\;{{tg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} + c^{*}} \right)}{b^{*} - c^{*}} \right\rbrack}}$where: ${e_{0}^{*} = \frac{a + b + c}{3}};{and}$${a^{*} = \frac{a}{e_{0}^{*}}};{b^{*} = \frac{b}{e_{0}^{*}}};{c^{*} = {\frac{c}{e_{0}^{*}}.}}$9. The method of claim 8 further including the step of identifying thefrequency of the second harmonic of the supply line voltages and usingthe frequency to identify a two phase supply voltage angle, the step ofidentifying the second harmonic components including identifying theq-axis second harmonic component by multiplying value A_(min) by thesine of the voltage angle and identifying the d-axis second harmoniccomponent by multiplying value A_(min) by the cosine of the voltageangle.
 10. The method of claim 5 wherein the step of using the referencevoltage to identify d and q-axis voltage difference values includesusing the reference voltage to identifying d and q-axis commandcurrents, obtaining d and q-axis feedback currents, subtracting the dand q-axis feedback currents from the d and q-axis command currents,respectively, and using the d and q-axis command currents to identifythe voltage difference values.
 11. The method of claim 2 wherein thecontroller receives a DC reference voltage and controls the converter tocause the DC voltage across the DC buses to track the reference voltage,the method further including the step of using the reference voltage toidentify first, second and third phase voltage difference values, thestep of identifying the second harmonic components including the step ofidentifying first, second and third phase components of the secondharmonic corresponding to the first, second and third supply lines,respectively, the step of modifying the command voltages includingmathematically combining the difference values and the second harmoniccomponents to identify first, second and third command voltages.
 12. Themethod of claim 11 wherein the step of mathematically combining includesadding the first, second and third phase second harmonic components tothe first, second and third phase difference values.
 13. The method ofclaim 11 wherein the step of identifying the peak amplitudes includesthe step of solving the following equations:$a = {{\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ca}_{-}{RMS}}^{2} - V_{{bc}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$$b = {{\sqrt{\frac{2}{3}} \cdot V_{{bc}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ab}_{-}{RMS}}^{2} - V_{{ca}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$${c = {{\sqrt{\frac{2}{3}} \cdot V_{{ca}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{bc}_{-}{RMS}}^{2} - V_{{ab}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}},$where a, b and c are the peak amplitudes and where Vab_RMS, Vbc_RMS andVca_RMS are the first, second and third RMS line-to-line voltages thatoccur between the first and second, second and third and third and firstlines, respectively.
 14. The method of claim 11 further including thestep of identifying the frequency of the supply line voltages and usingthe frequency to identify a supply voltage angle, the step ofidentifying the second harmonic components including identifying theq-axis second harmonic component by multiplying peak values a, b and cby the sine of the supply voltage angle, the sine of the supply voltageangle less 120 degrees and the sine of the supply voltage angle plus 120degrees, respectively.
 15. The method of claim 11 wherein the step ofusing the reference voltage to identify voltage difference valuesincludes using the reference voltage to identifying first, second andthird phase reference currents, obtaining first, second and third phasefeedback currents, subtracting the first, second and third phasefeedback currents from the first, second and third reference currents toidentify first, second and third command currents, respectively, andusing the first, second and third phase command currents to identify thevoltage difference values.
 16. An apparatus for use with a converterwhere the converter is actively controlled to convert three phasevoltages on three supply lines to a DC voltage across positive andnegative DC buses, the apparatus comprising: a processor programmed toperform the steps of: identifying the peak amplitudes of the three phasesupply line voltages; using the peak amplitudes to identify a secondharmonic component that would be generated on the supply lines by theconverter during normal operation due to unbalance in the peakamplitudes; and altering control of the converter as a function of theidentified second harmonic.
 17. The apparatus of claim 16 wherein theprocessor is further programmed to generate command voltages to controlthe converter, the processor performing the step of altering control ofthe converter by modifying the command voltages as a function of theidentified second harmonic.
 18. The apparatus of claim 17 wherein theprocessor performs the step of identifying the peak amplitudes bysensing the RMS line-to-line voltages and using the RMS line-to-linevoltages to identify the peak amplitudes.
 19. The apparatus of claim 18wherein the processor performs the step of identifying the peakamplitudes by solving the following equations:$a = {{\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ca}_{-}{RMS}}^{2} - V_{{bc}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$$b = {{\sqrt{\frac{2}{3}} \cdot V_{{bc}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ab}_{-}{RMS}}^{2} - V_{{ca}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$${c = {{\sqrt{\frac{2}{3}} \cdot V_{{ca}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{bc}_{-}{RMS}}^{2} - V_{{ab}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}},$where a, b and c are the peak amplitudes and where Vab_RMS, Vbc_RMS andVca_RMS are the first, second and third RMS line-to-line voltages thatoccur between the first and second, second and third and third and firstlines, respectively.
 20. The apparatus of claim 17 wherein the processorreceives a DC reference voltage and controls the converter to cause theDC voltage across the DC buses to track the reference voltage, theprocessor further programmed to perform the step of using the referencevoltage to identify d and q-axis voltage difference values, theprocessor performing the step of identifying the second harmoniccomponents by identifying d and q-axis components of the secondharmonic, the processor by mathematically combining the differencevalues and the second harmonic components to identify two phase d andq-axis command voltages.
 21. The apparatus of claim 20 wherein theprocessor is programmed to perform the step of mathematically combiningby adding the d and q-axis second harmonic components to the d andq-axis difference values.
 22. The apparatus of claim 20 wherein theprocessor is further programmed to perform the step of using the peakamplitudes to identify a DC offset and wherein the processor isprogrammed to modify the command voltages by adding the DC offset to theq-axis difference value along with the q-axis second harmonic component.23. The apparatus of claim 21 wherein the peak amplitudes of the threesupply line voltages are a, b and c and wherein the processor performsthe step of identifying the d and q axis second harmonic components byusing the peak amplitudes to identify a two phase amplitude A_(min) bysolving the following equation:A _(min) =A _(α)·Sin(φ_(α))+A _(β)·Cos(φ_(β))/2where:A _(α) =e* ₀/3·√{square root over ((a*+b*)²+(a*+c*)²+2(a*)²−b*·c*)}{square root over ((a*+b*)²+(a*+c*)²+2(a*)² −b*·c*)}{square rootover ((a*+b*)²+(a*+c*)²+2(a*)² −b*·c*)}A _(β) =e* ₀/√{square root over (3)}·√{square root over ((b*+c*)²−b*·c*)}φ_(α)=arctg [√{square root over (3)}·(b*−c*)/4a*+b*+c*]φ_(β)=arctg [√{square root over (3)}·(b*−c*)/b*−c*]where:e* ₀ =a+b+c/3; anda*=a/e* ₀ ; b*=b/e* ₀ ; c*=c/e* ₀.
 24. The apparatus of claim 23 whereinthe processor is further programmed to perform the steps of identifyingthe frequency of the second harmonic of the supply line voltages andusing the frequency to identify a two phase supply voltage, theprocessor performing the step of identifying the second harmoniccomponents including identifying the q-axis second harmonic component bymultiplying value A_(min) by the sine of the voltage angle andidentifying the d-axis second harmonic component by multiplying valueA_(min) by the cosine of the voltage angle.
 25. The apparatus of claim23 wherein the processor performs the step of using the referencevoltage to identify d and q-axis voltage difference values using thereference voltage to identifying d and q-axis command currents,obtaining d and q-axis feedback currents, subtracting the d and q-axisfeedback currents from the d and q-axis command currents, respectively,and using the d and q-axis command currents to identify the voltagedifference values.
 26. The apparatus of claim 17 wherein the processorreceives a DC reference voltage and controls the converter to cause theDC voltage across the DC buses to track the reference voltage, theprocessor further programmed to perform the step of using the referencevoltage to identify first, second and third phase voltage differencevalues, the processor performing the step of identifying the secondharmonic components by identifying first, second and third phasecomponents of the second harmonic corresponding to the first, second andthird supply lines, respectively, the processor programmed to performthe step of modifying the command voltages by mathematically combiningthe difference values and the second harmonic components to identifyfirst, second and third command voltages.
 27. The apparatus of claim 26wherein the processor is programmed to perform the step ofmathematically combining by adding the first, second and third phasesecond harmonic components to the first, second and third phasedifference values.
 28. The apparatus of claim 26 wherein the processoris programmed to identifying the peak amplitudes by solving thefollowing equations:$a = {{\sqrt{\frac{2}{3}} \cdot V_{{ab}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ca}_{-}{RMS}}^{2} - V_{{bc}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$$b = {{\sqrt{\frac{2}{3}} \cdot V_{{bc}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{ab}_{-}{RMS}}^{2} - V_{{ca}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}$${c = {{\sqrt{\frac{2}{3}} \cdot V_{{ca}_{-}{RMS}}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{{bc}_{-}{RMS}}^{2} - V_{{ab}_{-}{RMS}}^{2}} \right)}{V_{{ab}_{-}{RMS}} + V_{{bc}_{-}{RMS}} + V_{{ca}_{-}{RMS}}}}}},$where a, b and c are the peak amplitudes and where Vab_RMS, Vbc_RMS andVca_RMS are the first, second and third RMS line-to-line voltages thatoccur between the first and second, second and third and third and firstlines, respectively.
 29. The apparatus of claim 26 wherein the processoris further programmed to perform the steps of identifying the frequencyof the supply line voltages and using the frequency to identify a supplyvoltage angle, the processor performing the step of identifying thesecond harmonic components by identifying the q-axis second harmoniccomponent by multiplying peak values a, b and c by the sine of thesupply voltage angle, the sine of the supply voltage angle less 120degrees and the sine of the supply voltage angle plus 120 degrees,respectively.
 30. The apparatus of claim 26 wherein the processor isprogrammed to perform the step of using the reference voltage toidentify voltage difference values by using the reference voltage toidentifying first, second and third phase reference currents, obtainingfirst, second and third phase feedback currents, subtracting the first,second and third phase feedback currents from the first, second andthird reference currents to identify first, second and third commandcurrents, respectively, and using the first, second and third phasecommand currents to identify the voltage difference values.
 31. A methodfor use with a controller and a converter wherein the controllerreceives a reference voltage and generates first, second and third phasecontrol voltages as a function of the reference voltage, the converterreceiving the first, second and third phase control voltages and first,second and third phase line voltages and converting the line voltages toa DC voltage across positive and negative DC buses as a function of thecontrol voltages where the line voltages may be unbalanced, the methodfor substantially reducing the second harmonics in the first, second andthird phase line currents caused by drawing current from the lines whenthe line voltages are unbalanced, the method comprising the steps of:identifying first, second and third RMS line-to-line voltages; using theRMS line-to-line voltages to identify peak line voltage values;mathematically combining the peak line voltage values and at least aderivative of the reference voltage to generate the first, second andthird phase command voltages; and using the first, second and thirdphase command voltages to control the converter.
 32. A method for usewith first, second and third voltage supply lines that feed a converter,the method for identifying the peak amplitudes of the line voltages, themethod comprising the steps of: obtaining RMS line-to-line voltages fromthe supply lines; and solving the following equations to identify thepeak amplitudes a, b and c of the voltages on the first, second andthird supply lines, respectively: $\begin{matrix}{a = {{\sqrt{\frac{2}{3}} \cdot V_{ab\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{ca\_ RMS}^{2} - V_{bc\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}} \\{b = {{\sqrt{\frac{2}{3}} \cdot V_{bc\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{ab\_ RMS}^{2} - V_{ca\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}} \\{c = {{\sqrt{\frac{2}{3}} \cdot V_{ca\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{bc\_ RMS}^{2} - V_{ab\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}}\end{matrix}$  where Vab_RMS, Vbc_RMS and Vca_RMS are the first, secondand third RMS line-to-line voltages that occur between the first andsecond, second and third and third and first lines, respectively. 33.The method of claim 14 also for identifying a second harmonic componentthat would be generated on the supply lines by the converter duringnormal operation due to unbalance in the peak amplitudes and using theidentified second harmonic component to alter control of a converterlinked to the supply lines to convert the AC voltage on the supply linesto a DC bus voltage across positive and negative DC buses.
 34. A methodfor use with a controller and a converter wherein the controllerreceives a reference voltage and generates first, second and third phasecontrol voltages as a function of the reference voltage, the converterreceiving the first, second and third phase control voltages and first,second and third phase line voltages and converting the line voltages toa DC voltage across positive and negative DC buses as a function of thecontrol voltages where the line voltages may be unbalanced, the methodfor substantially reducing second harmonics in the first, second andthird phase line currents caused by drawing current from the lines whenthe line voltages are unbalanced, the method comprising the steps of:obtaining first, second and third RMS line-to-line voltages; identifyingthe first, second and third peak amplitudes a, b and c, of the first,second and third supply line voltages, respectively, by solving thefollowing equations: $\begin{matrix}{a = {{\sqrt{\frac{2}{3}} \cdot V_{ab\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{ca\_ RMS}^{2} - V_{bc\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}} \\{b = {{\sqrt{\frac{2}{3}} \cdot V_{bc\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{ab\_ RMS}^{2} - V_{ca\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}} \\{{c = {{\sqrt{\frac{2}{3}} \cdot V_{ca\_ RMS}} + {\sqrt{\frac{3}{2}} \cdot \frac{\left( {V_{bc\_ RMS}^{2} - V_{ab\_ RMS}^{2}} \right)}{V_{ab\_ RMS} + V_{bc\_ RMS} + V_{ca\_ RMS}}}}},}\end{matrix}$  respectively, where Vab_RMS Vbc_RMS and Vca_RMS are thefirst, second and third RMS line-to-line voltages that occur between thefirst and second, second and third and third and first lines,respectively; using the peak amplitudes to identify feed forwardvoltages; mathematically combining the feed forward voltages and atleast a derivative of the reference voltage to generate the first,second and third phase command voltages; and using the first, second andthird phase command voltages to control the converter.
 35. A method foruse with a controller and a converter where the controller activelycontrols the converter to convert three phase voltages on three supplylines to a DC voltage across positive and negative DC buses, the methodcomprising the steps of: identifying the peak amplitudes a, b and c ofthe three phase supply line voltages; identifying a two phase amplitudevalue A_(min) by solving the following equation: $\begin{matrix}{A_{\min} = \frac{{A_{\alpha} \cdot {{Sin}\left( \varphi_{\alpha} \right)}} + {A_{\beta} \cdot {{Cos}\left( \varphi_{\beta} \right)}}}{2}} \\{{where}\text{:}} \\{A_{\alpha} = {\frac{e_{0}^{*}}{3}\sqrt{\left( {a^{*} + b^{*}} \right)^{2} + \left( {a^{*} + c^{*}} \right)^{2} + {2\left( a^{*} \right)^{2}} - {b^{*} \cdot c^{*}}}}} \\{A_{\beta} = {\frac{e_{0}^{*}}{\sqrt{3}} \cdot \sqrt{\left( {b^{*} + c^{*}} \right)^{2} - {b^{*} \cdot c^{*}}}}} \\{\varphi_{\alpha} = {{arc}\;{{tg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} - c^{*}} \right)}{{4a^{*}} + b^{*} + c^{*}} \right\rbrack}}} \\{\varphi_{\beta} = {{arc}\;{{tg}\left\lbrack \frac{\sqrt{3} \cdot \left( {b^{*} + c^{*}} \right)}{b^{*} - c^{*}} \right\rbrack}}} \\{{where}\text{:}} \\{{e_{0}^{*} = \frac{a + b + c}{3}};{and}} \\\begin{matrix}{{a^{*} = \frac{a}{e_{0}^{*}}};} & {{b^{*} = \frac{b}{e_{0}^{*}}};} & {c^{*} = \frac{c}{e_{0}^{*}}}\end{matrix}\end{matrix}$ using value Amin to identify d and q-axis second harmoniccomponents that would be generated on the supply lines by the converterduring normal operation due to unbalance in the peak amplitudes; usingthe d and q-axis second harmonic components to alter control of theconverter thereby substantially minimizing the second harmoniccomponents on the three phase supply lines.
 36. A method for use with acontroller and a converter where the controller actively controls theconverter to convert three phase voltages on three supply lines to a DCvoltage across positive and negative DC buses, the method comprising thesteps of: identifying unbalance in the peak amplitudes of the threephase supply line voltages; and using the unbalance to alter control ofthe converter to substantially eliminate generation of second harmonicson the supply lines due to active converter control.